ANALYSIS OF ELECTRON TRANSPORT THROUGH NOVEL
NANOELECTRONIC AND SPINTRONIC DEVICES
A THESIS
SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE
MASTER OF SCIENCE
BY
JAMES B. CUTRIGHT
ADVISOR: DR. ERIC HEDIN
BALL STATE UNIVERSITY
MUNCIE, INDIANA
JULY 2012
i
I would like to dedicate this work to my mother,
Lori C. Cutright:
She inspired my sense of scientific
curiosity at a young age,
and has unwaveringly supported the
entirety of my academic career.
ii
ABSTRACT
Traditionally, computer components like transistors have been large enough that
they obey classical physical law. Today, as computers continue to be miniaturized,
traditional components no longer operate correctly because their dimensions are close to,
or on, the atomic scale. Quantum physics takes over in this regime, so new components
and computing techniques must be devised.
Two such new sets of devices and techniques are nanoelectronics and spintronics.
Nanoelectronics are very similar to current microelectronics, but their size constraints
force them to operate in the quantum mechanical regime. Spintronics is a field whereby
the intrinsic fermion spin of the electron is used to represent binary bits. In my work we
propose new device schemes with applications in both nanoelectronics and spintronics.
In the nanoelectronic realm we propose that a series of Aharonov-Bohm Rings,
with quantum dots in each arm of the rings, can be linked via intermediate quantum dots.
Applying an external magnetic field to this system proves interesting, since the system
can be made to change its overall conductivity via this tunable, external magnetic field.
Typical Single Electron Transistors (SET) operate by employing the Coulomb Blockade
Effect and Voltage Biasing to manipulate the conductivity of a nanostructure . The
proposed device would, in principle, perform the same task, but the novel approach may
have certain advantages over the SET.
In the field of spintronics we propose a simple, spin-polarizing device. Typically,
any current used by a computer has electrons with randomly oriented spin-states. This is
okay, however, because the system ignores these states. A spintronic device, however,
iii
would need an input current with a precise spin orientation, so that the spin-states of the
electrons could be accurately manipulated. This means a current that will be fed into a
spintronic device needs to be filtered first, to ensure that the device has a base current to
work with. While spin-state filtering has been researched and achieved experimentally it
is inefficient, often only reaching thirty or forty percent polarization. Other techniques
have been developed that do reach up to full spin-filtering of a current, but these are
expensive and time consuming since they generally require extensive lithographic work
and high grade graphene sheets or carbon nanotubes. We endeavor to show that a simple
series of QDs can potentially be used to efficiently spin-polarize a current, with the added
bonus that current-day techniques which are well understood could be used to construct
such a device.
Each system will be attached to input and output nanowires in which the g-factor
is close to zero, so that any external magnetic field penetrating the device will not affect
the output in the transmission lines themselves. Each system is then modeled with the
Tight Binding Approximation (TBA) to the Schrödinger Equation. Mathematica is then
used to solve the systems of equations generated with TBA, so that the transmission of
the system can be plotted as a function of electron energy and external magnetic field. It
is found that a series of Aharonov-Bohm rings in series may be useful as a spin-polarizer,
and potentially as a single electron transistor. The quantum dot spin-polarizer may also be
used to spin-polarize currents, but in combination with multiple Aharonov-Bohm rings, a
stronger spin-polarized current may be produced.
iv
ACKNOWLEDGMENTS
I would first like to thank Dr. Eric Hedin for all of his hard work, encouragement,
and patience while he worked with me for the past two years. He is an excellent advisor
and mentor, and I am thankful he allowed me to work with him. I would like to thank Dr.
Yong Joe as well, for all of the mentorship he provided while I worked with Dr. Hedin,
and for teaching me about an incredibly broad range of topics in nanoscience. For all of
the hard work that he put into his classes, everything that I learned from him about
physics and about teaching, his excellence as a Professor, and his sense of humor, I thank
Dr. Antonio Cancio.
For all of their hard work, dedication, and time spent advising me, I would like to
thank my committee members: Dr. Hedin, Dr. Cancio, Dr. Joe, and Dr. Robertson. All of
the time and energy spent helping me to prepare these past two years is much
appreciated.
For all the long hours spent on homework, the homework help, and all the
camaraderie, I thank my fellow graduate students.
Finally, I would like to thank the Ball State University Physics and Astronomy at
large; I learned and understood far more than I expected I could with their help, and I was
made to feel very welcome.
v
TABLE OF CONTENTS
Page
DEDICATION ..................................................................................................................... i
ABSTRACT ........................................................................................................................ ii
ACKNOWLEDGEMENTS ............................................................................................... vi
LIST OF FIGURES ........................................................................................................... iv
CHAPTERS
1. INTRODUCTION ...................................................................................................1
1.1
Spintronics .......................................................................................1
1.2
Semiconductors and Quantum Dots.................................................8
1.3
Zeeman Splitting ............................................................................15
1.4
System Designs ..............................................................................21
2. METHODS OF ANALYSIS .................................................................................26
2.1.
Energy Scales .................................................................................26
2.2.
The Tight Binding Approximation ................................................27
2.3.
Weighted Spin-Polarization ...........................................................30
2.4.
Calculation of I-V Curves ..............................................................32
3. THE AHARONOV-BOHM RING ........................................................................33
3.1
Simulation Parameters ...................................................................33
3.2
The Multi-Aharonov-Bohm Ring System .....................................35
3.3
Applications ...................................................................................50
4. THE QUANTUM DOT SPIN-POLARIZER ........................................................52
4.1
Simulation Parameters ...................................................................52
4.2
The Double Quantum Dot System .................................................53
4.3
The Triple Quantum Dot System ...................................................56
4.4
Applications ...................................................................................60
5. FUTURE WORK ...................................................................................................61
6. SUMMARY AND CONCLUSION ......................................................................62
7. REFERENCE .........................................................................................................64
8. APPENDIX A ........................................................................................................67
9. APPENDIX B ........................................................................................................75
10. APPENDIX C ........................................................................................................76
11. APPENDIX D ........................................................................................................77
12. APPENDIX E ........................................................................................................78
vi
LIST OF FIGURES
Figure 1.1.1
Plot of Moore’s Law ....................................................................................2
Figure 1.1.2
Comparison of spintronic and microelectronic devices ..............................5
Figure 1.1.3
Experimental set up of a spin-filter .............................................................6
Figure 1.2.1
a) E vs. k example of electrons in a semiconductor. b) E vs. k for
electrons in a vacuum and GaAs ..................................................................8
Figure 1.2.2
Sample band diagrams of Si and GaAS ....................................................10
Figure 1.2.3
Theory of heterostructures formed by two dissimilar semiconductors .....10
Figure 1.2.4
Diagrams of the potential well formed at a heterojunction .......................11
Figure 1.2.5
Formation of the AlGaAs/GaAs/AlGaAs heterostructure .........................13
Figure 1.2.6
A quantum dot formed using quantum point contacts...............................13
Figure 1.2.7
A quantum dot formed by an AlGaAs/GaAs/AlGaAs heterostructure .....14
Figure 1.2.8
Quantum wire formed by AlGaAs/GaAs/AlGaAs heterostructure ...........15
Figure 1.3.1
Electron magnetic moments aligning and anti-aligning to an external
magnetic field.............................................................................................18
Figure 1.3.2
Simple E vs. x plot of a quantum dot formed by a double potential
barrier .........................................................................................................19
Figure 1.3.3
Schematic of a single quantum dot ............................................................19
Figure 1.3.4
T vs. E of the single quantum dot with no ZSE (Fig. 1.3.3) ......................20
Figure 1.3.5
T vs. E of the single quantum dot with ZSE (Fig. 1.3.3) ...........................20
Figure 1.4.1
Schematic of a single Aharonov-Bohm ring .............................................22
Figure 1.4.2
Physical example of an Aharonov-Bohm ring ..........................................22
Figure 1.4.3
Schematic of the double Aharonov-Bohm ring without spin-split
IQDs ..........................................................................................................23
Figure 1.4.4
Schematic of the Aharonov-Bohm ring system with spin-split
IQDs and an n number of rings ..................................................................24
Figure 1.4.5
Schematic of the double quantum dot spin-polarizer ................................24
Figure 1.4.6
Physical construction of quantum dots in series........................................25
Figure 2.2.1
Plot of a periodic potential, or the “Dirac Comb” .....................................28
Figure 3.1.1
Schematic of the double Aharonov-Bohm ring with applied external
magnetic field............................................................................................35
Figure 3.2.1
T vs. E through an Aharonov-Bohm ring without an applied external
magnetic field............................................................................................37
vii
Figure 3.2.2
T vs. E through an Aharonov-Bohm ring with an applied external
magnetic field............................................................................................37
Figure 3.2.3
Transmission through the un-split IQD multi-ring system for two
(a-c), five (d-f), ten (g-i), and twenty (j-l) rings........................................38
Figure 3.2.4
Plot of T for the multi-ring system with changing ZSE. ..........................40
Figure 3.2.5
T vs. E for the contour plot in Fig. 3.2.4 ...................................................40
Figure 3.2.6
Transmission through the multi-ring system with spin-split IQDs for
for two (a-c), five (d-f), ten (g-i), and twenty (j-l) rings ...........................42
Figure 3.2.7
Plot of the T vs. ZSE vs. E for the five ring system ..................................44
Figure 3.2.8
T vs. E cuts for the contour plot in Fig. 3.2.7............................................44
Figure 3.2.9
I-V curves accompanying the T vs. E curves in Fig. 3.2.6 .......................46
Figure 3.2.10 Plot of the ΔPol vs. ZSE vs. E for the five ring system .............................48
Figure 3.2.11 ΔPol vs. E plots accompanying Fig. 3.2.10 ...............................................48
Figure 3.2.12 T vs. ZSE vs. E for the five ring system with V = 0.3 ...............................49
Figure 3.2.13 T vs. E plots accompanying Fig. 3.2.12 ....................................................49
Figure 3.2.14 Contour plot of ΔPol vs. ZSE vs. E for the five ring system with
V = 0.3 (Fig. 3.2.11, 3.2.12) .......................................................................50
Figure 4.1.1
Schematic of the double QD system .........................................................53
Figure 4.2.1
T vs. E for the DQDS ................................................................................54
Figure 4.2.2
T vs. E when the FQD site energy equals that of one of the Zeeman
split energy levels .....................................................................................55
Figure 4.2.3
Optimization of the basic DQDS ...............................................................55
Figure 4.3.1
TQDS T vs. E corresponding to that of the DQDS in Fig. 4.2.2 ..............56
Figure 4.3.2
T vs. E plots for the TQDS corresponding to Fig. 4.2.1, so that the
DQDS and TQDS may be compared ........................................................57
Figure 4.3.3
Optimization of TQDS with parameters equal to those in Fig. 4.2.3 ........58
Figure 4.3.4
Optimization of TQDS where V = 1.3, ZSE = 1.9, and the site energy
for the FQD is ƐQD = 1.9 ..........................................................................58
Figure 4.3.5
Contour plot of T vs. V vs. E, where ZSE = 1.9. This plot represents
the max transmission for the system, but not the max ΔPol .....................59
Figure 4.3.6
Plots showing ΔPol of the T from Fig. 3.4.5 for a) V = 1.3,
and b) V = 0.55 ..........................................................................................59
Chapter 1: Introduction
1.1. Spintronics
Spintronics is a portmanteau of “Spin Electronics”, and is the study of active
control and manipulation of spin degrees of freedom in solid state systems for use in
computation or study [1]. The typical circuit element in a modern computer specifically
takes advantage of the “charge degree of freedom” of the electron [2]. This means that a
bit of information is represented by the presence or absence of charge in a solid state
system, like a MOSFET. Spintronic systems would take advantage, instead, of the spin
degrees of freedom of the electron.
The primary impetus behind spintronic research is Moore’s Law (Fig. 1.1.1).
Moore’s law states that microprocessor computing power will double every 18 months,
as silicon chips become smaller [3,4].
2
Figure 1.1.1: A graph showing Moore's Law. Every 18 months the processing power
of computers doubles as chips shrink[4].
This law, however, does not take into account the fact that once the silicon
devices we use today are on the size scale of atoms, they cease to function. Even worse,
these same silicon devices experience problems with leakage and heat dissipation long
before they get so small. This means that the devices will have difficulty confining the
electrons they need to do calculations and the heat generated by the devices will be great
enough to damage the unit.
Current silicon-based chips have already hit this size-limiting problem; many
people are trying to design novel devices that can perform computations which are so
small that they operate at the atomic level, or the nanometer scale. The single electron
transistor is one way around this problem, and is discussed later. It operates using
electron charge, like a traditional microelectronic device, but also incorporates quantum
3
theory, so that it can function at the nanometer scale. Even so, spintronic devices may
provide an equally good or better solution, and there are certainly advantages to
spintronics.
Spintronics has a few advantages that seem to make it a favorable solution for
making even smaller transistors [1]. First, is the enormous amount of information that
could be theoretically held and processed by spin-states, through the use of qubits and the
quantum entanglement of qubits.
Another big advantage to spintronics is the ability to control spin with magnetic
and electric fields. Spin-states can be controlled by magnetic fields since spin is just a
manifestation of a magnetic dipole moment. Electric fields can also be used to control a
spin-state through things like the Rashba field effect [5], where the spin-state is made to
flip because of an external electric field. Even though we can use both magnetic and
electric fields to exert control over the electron’s spin-degrees of freedom, we will
concentrate on how the magnetic control can be of use.
Still another advantage of spintronics over traditional electronics is spincoherence. The spin-states of an electron have a very long relaxation time [1]. It is also
difficult to destroy a spin-state with impurities in the transmitting medium or electronelectron interactions. This means that spin-states of electrons provide a very robust
method for handling information.
The last reason that spintronics have gained so much interest is centered on Shor’s
and Grover’s algorithms [3]. These algorithms perform basic functions; factorization of
large prime numbers and the inversion of a function, respectively. Quantum computing
4
has the potential to increase the processing speed of these two important functions
dramatically, making computers not only smaller, but even faster than previously
thought.
Overall, the advantages of the spintronic quantum computer are quite impressive.
Despite some of the shortcomings, like measuring the spin-states, many devices are being
designed that take advantage of spin, and some are even already on the market.
Spintronic Devices
Many of the current spintronic devices depend on two phenomena; giant
magnetoresistivity [6, 7] and electron transport through quantum dots [8]. Many other
phenomena have been championed as well, but GMR is quite prevalent as it is already
used in modern hard drives, and quantum dot structures hold promise as they may be
easily integrated into current computing devices. Both GMR and electron transport
through QDs take advantage of the fact that the two spin-states of an electron will react to
potential barriers or material interfaces in different ways. Our work is centered on the
transport through quantum dots and how it can be potentially used to create a spintransistor and a spin-polarizer.
The main goal behind most spintronic research is to create spintronic analogs to
current digital devices. In Fig. 1.1.2 is a short outline of the standard electronic devices
and the spintronic analogs that we are trying to, or have already, created. Notice that
standard electronic devices have been “unified”. This means that they all work on the
same principle; all of them are made of semiconducting materials, and employ some sort
5
of junction to process a current or voltage. This unification also makes standard
electronic devices easy to mass produce and to integrate with one another, since they all
function on the same basic principles. Spintronics are not unified, as they employ many
phenomena, since we are still trying to develop the basic devices. Below are descriptions
of two spintronic devices that have already been built and tested.
Standard (Electronic) Devices (unified)
Spintronic Devices (not unified)
Traditional Hard Disk
Modern Hard Disk
ACDC Rectifiers
Spin-Filters/Batteries
Diodes
Spin-Diodes
Transistors
Spin-Transistors
Amplifiers
Spin-Amplifiers
TFigure 1.1.2: Electronics devices and their spintronic analogs.
Spin-Filters
A spin-filter is used to take a spin-unpolarized current and remove one spin-state,
so that only one remains. This is often accomplished by splitting the energy levels in a
QD and then somehow isolating one of the resonant Zeeman split energy levels. In Fig.
6
1.1.3 is a circuit that is designed to take advantage of the resonant energies of a Zeeman
split QD.
Figure 1.1.3: Experimental setup of a proposed spin filter, where the space next to
“E” is a quantum dot cut out of the 2DEG by the confining gate voltages. [9].
The unfortunate aspect to this particular spin-filter is that it can reach only 70%
spin-polarization, at maximum. Typically only about 50% is actually achieved [9].
Currently, other spin-polarizers that have been demonstrated do not polarize a current to a
very high degree either. This is quite bad for spintronics, since all computational devices
require that the baseline current going into the system be very homogenous. For an
electronic circuit this means rectifying alternating current into a usable direct current, and
for a spintronic device rectifying a direct current further into a spin-polarized current.
7
Later, a potential design schematic for a simple spin-polarizer is suggested and
subsequent simulation results are shown which provide a basic proof of principle.
Overview
To date, one of the biggest areas of research in spintronics is the production of
spin-polarized current. While there are systems that are capable of producing this, the
systems are still too large, and need to be further refined so that the overall spinpolarization of the generated current is purer. Spin-amplification is still a hot topic
because a method for amplifying a spin-polarized current has yet to be rigorously
demonstrated. Before any quantum computer can be built we still need to develop clever
new algorithms that operate on quantum principles, since we have only Shor’s and
Grover’s algorithms to work with. Graphene spintronics have recently shown a lot of
promise and are being heavily researched [10, 11]. Whether or not this will provide
anything useful is still up in the air. There have been some real triumphs, however, in
memory. To date many computers now use spintronic-based hard drives that use GMR as
the primary method of reading and writing information. These hard drives can store close
to an order of magnitude more information than their counterparts. Along with this, some
companies have also developed spintronic magnetic random access memory with read
write times as fast as 50 nanoseconds; a significant improvement [12]. In the end, there
is still a lot of work to be done before we can start to unify the methods for creating
spintronic devices and then start building usable computers.
8
1.2. Semiconductors and the Formation of Nanoscale Devices
A semiconductor is a specific type of material that traditionally is crystalline in
nature, like silicon, gallium arsenide, or germanium, all of which are very prevalent in
modern micro and nanoelectronics. At absolute zero they have the property of being very
good insulators, because their band structure has an intrinsic energy gap between the
filled valence band and the empty conduction band (Fig. 1.2.1).
Figure 1.2.1: (a) Example curves of the conduction and valence bands in a
semiconductor. (b) Energy versus wavevector plot for an electron in vacuum and
GaAs [8].
This gap is often referred to as a forbidden zone, since an electron wave function cannot
exist there. This means that electrons will tend to stay in the filled valence shells, unless
they gain energy equal to or above this energy gap. By doping these materials with
impurities, forming heterojunctions and structures with them, and applying external fields
to them, we can exert a certain amount of control over the electrons at the valence band
edge of the semiconductor structures. Under the right conditions, an electron can be given
enough energy to jump past the forbidden zone and into the conduction band, so that we
9
can move the electron around in a predictable and useful way. Likewise, we can also
simply leave the electron in the valance band. This level of control over the relative
conductivity of the electrons in the structure makes these materials invaluable, as they
can be used to create transistors, switches, diodes, photovoltaic cells, many kinds of
detectors, and thermistors [13].
The band gap of a semiconductor is defined as the energy difference between the
lowest point in the conduction band, or the conduction band edge, and the highest point
in the valence band, or the valence band edge. At absolute zero, the thermal energy of the
electrons is not high enough to excite electrons into the conduction band, and the material
is an insulator. If the material is at room temperature, then some of the electrons may be
energetic enough to jump across the band gap. When this happens, however, the valance
orbital of one of the atoms in the material is now left unfilled. This electrical hole left in
the orbital can be treated as a particle with charge equal to the absolute value of the
electron charge. This vacant part of an orbital will react like a particle in all ways; it will
respond to electric and magnetic fields; its overall wave vector will cancel out that of the
electron that has been excited, meaning there will be no net change in the wave vector of
the system, which is good because momentum is conserved (Fig. 1.2.2).
10
Figure 1.2.2: Sample band diagrams of Silicon and GaAs, showing where the
electrons and their holes tend to arrange themselves [14].
Different semiconducting materials will have different band gaps relative to each other.
The Fermi energy, EF, of a semiconductor is related to its band gap, and a material with a
large band gap will have a higher EF than that of one with a smaller band gap. This
property is useful, since it is what allows us to create heterojunctions (Fig. 1.2.3).
Figure 1.2.3: A heterojunction formed by two dissimilar semiconductors. The E vs k
plots show that the electron motion is limited to one direction [15].
A heterojunction is formed when two different semiconductors are layered one on
top of the other. Layering even more semiconductors onto a system creates a
11
heterostructure. The difficulty in creating these systems, however, is that at the boundary
between two dissimilar semiconductors a discontinuity is formed in either the valence or
conduction band. This discontinuity creates a structural strain on the device. This is why
certain heterojunctions are more prevalent than others. The best types of semiconductors
for forming these junctions are III-V, and their solid solution counterparts. A good
example of this is the AlGaAs/GaAs/AlGaAs heterostructure, where AlGaAs is a solid
solution of GaAs. The lattice constant of GaAs is 5.6533 Å, and that of the ternary
compound AlxGa1-xAs can have a lattice mismatch of less than 0.1% [14]. This means
that the AlGaAs/GaAs/AlGaAs heterostructure has an approximately constant lattice
constant [8], making the material more stable, and thus suitable for device construction.
At the boundary of dissimilar semiconductors another interesting effect takes
hold, if we use modulation doping to add impurities to one of the materials. The AlGaAs
layer in Fig. 1.2.4 has been doped such that it is now an n-type semiconductor, meaning
that it has extra donor electrons that it can lose, without sacrificing the completeness of
its electron orbitals.
Figure 1.2.4: Diagrams showing how the donor electrons are redistributed when two
dissimilar semiconductors form a heterojunction, and the subsequent potential well
created. The new potential well confines the donor electrons in a 2DEG [15].
12
The EF of the AlGaAs is higher than that of the GaAs layer. Looking to lower their
energy, the donor electrons will tend to migrate from the higher energy state, to the GaAs
layer. This in turn lowers the EF of the AlGaAs layer, and raises the EF of the GaAs layer
until they are equal to each other. At the boundary, however, the mismatch in energy will
remain, even as the overall EF equalizes. This means that as the electrons migrate to the
GaAs they will be unable to return to their donors, even after the EF have equalized.
Furthermore, the donor electrons lost energy ΔEC, while the EF of the GaAs has
increased. By rearranging these conduction bands a triangular potential well is formed
and a large number of electrons at the barrier, forming a 2D electron gas (2DEG). For this
reason we refer to the heterojunction as a low-dimensional system, since the electrons
can’t move in all three directions. The carrier concentration in this region will be quite
high, meaning that the electrons are highly mobile. Overall, confining the electrons and
making a 2DEG makes the electron quite easy to manipulate via external fields, making
heterojunctions excellent for device fabrication.
When building devices, it is often more useful to confine the electrons in a thin
layer of GaAs that is sandwiched between two thicker layers of AlGaAs. This structure
effectively forms a potential well barrier in the GaAs layer, and forces the electrons to
travel only in the plane of the GaAs layer (Fig. 1.2.5). This high level of confinement
induces electron energy quantization, and allows us to build structures that can
manipulate electrons at the nanoscale in predictable and useful ways, such as quantum
dots and quantum wires.
13
Figure 1.2.5: Formation of an AlGaAs/GaAs/AlGaAs heterostructure [8].
Using lithographic techniques quantum dots (QD) and quantum wires can be
formed in heterostructures. One such method uses quantum point contacts (QPC) to apply
an electric field perpendicular to the 2DEG (Fig. 1.2.6). This cuts the 2DEG into shapes
that are then useable, and can be used to further confine electrons into even smaller
regions.
Figure 1.2.6: A quantum dot formed by quantum point contacts emitting an electric
field into the 2DEG, providing further confinement [15].
14
In one case we can confine the electrons to move in only one direction, creating a
quantum wire. By further confining the electrons we create a quantum dot. These
structures are small enough that they have a discrete, measurable energy spectrum, like
an atom. For this reason we often refer to QDs as artificial atoms, and arrays of them as
artificial molecules. We can also build QDs by growing them directly on the substrate,
where the substrate has been chemically etched to force the growth of the QDs in only
certain regions (Fig. 1.2.7). This method is also useful for building quantum wires (Fig.
1.2.8), but it is often difficult to control precisely, and requires great knowledge about the
growth processes of these complex materials, as well as how the electrons are transported
in them. Overall, by building systems of quantum wires and dots we can begin to build
devices, such as transistors, out of them, but on the nanoscale. This has many
technological implications, since the potential to scale our computational devices down
by factors of 100 or more means much greater processing power than is currently
available [16].
Figure 1.2.7: A quantum dot formed by confining the electron to a small area in a
semiconducting heterostructure [8].
15
Figure 1.2.8: A quantum wire formed from a thin heterostructure. The extra
confinement in the x direction forces the electron to travel only in the y direction [8].
1.3. Zeeman Splitting
The Zeeman Effect is a phenomenon whereby the application of an external
magnetic field to a particular electron energy level causes that energy level to split into
two distinct states [5]. Electrons with differing quantum numbers, n, l, and m, may in
fact have the same overall energy. When this happens we say that the electron wave
functions at this energy are degenerate. The electron, being a fermion, has an intrinsic
magnetic dipole moment, μj. This magnetic moment can align or anti-align with an
external magnetic field. Electrons with the same energy, but different quantum numbers,
tend to react to an external perturbation, like a magnetic field, differently, such that the
degeneracy in the system is lifted. In general, an electron energy level will have pairs of
electrons, and with the application of an external magnetic field one of the electrons will
be polarized spin up and the other spin down. Whenever we apply an external magnetic
field, the electrons in these pairs will tend to be energetically split. Typically, the spin up
16
electron will gain energy and the spin down will lose, although this is not the case in
every material.
The total angular momentum of an electron can be written as,
⃗⃗⃗
⃗⃗⃗
⃗⃗⃗ ,
(1.3.1)
where the total angular momentum, , is simply the combination of the total orbital
angular momentum, ⃗⃗⃗ , and the spin orbital angular momentum, ⃗⃗⃗
also referred to as
the magnetic moment of the electron [5]. Historically, the orbital angular momentum was
first recognized and manipulated with external fields. Later, via the Stern-Gerlach
experiment it was found that electrons also possess an intrinsic spin. The effect that
magnetic fields can have on these spin states was not well understood at first, so it is
referred to as the anomalous Zeeman Effect [17]. In general, this work focuses on the
anomalous Zeeman Effect, since the orbital angular momentum of an electron entering a
quantum dot is not affected by the application of an external magnetic field [18]. This
occurs because the electrons are being confined to a 2DEG. An electron will feel the
magnetic field via the Lorentz force,
⃗.
(1.3.2)
Typically the magnetic field in the simulations, however, will be parallel to the
heterostructures forming the substrate for the quantum dots. This means that the force felt
by the electrons will tend to push them against the barrier that they are being confined
with, so their orbital degrees of freedom are limited in the QD. This simplifies things
greatly, since we can focus only on the splitting of the spin-states of the electrons, and
17
effectively ignore the orbital motions of the electrons. This means that the total angular
momentum is ⃗⃗⃗
The magnetic moment, ⃗⃗⃗ , of an electron, or any charged particle, in free space is
defined as:
⃗⃗⃗
where
,
(1.3.3)
is the total angular momentum of the particle, and γ is the “gyromagnetic ratio”
or “magnetogyric ratio”; the ratio of the magnetic moment of the particle to its angular
momentum [13]. It is useful to define another quantity, the g-factor, as well, which is a
dimensionless constant that describes the charge to mass distribution of an electron:
(1.3.4)
where
is the Bohr Magneton. Any electron in an external magnetic
field, then, when it aligns or anti-aligns with the field it will have energy:
⃗⃗⃗⃗
⃗
(1.3.5)
In terms of the g-factor this energy is:
,
(1.3.6)
where mj is the azimuthal quantum number. In this case, however, since we can neglect
the orbital angular momentum,
The g-factor of an electron plays a large role in
how the two potential spin orientations will align with the magnetic field. For a free
electron, g is 2.0023, which we usually just approximate as 2 [13]. In these systems,
however, the electrons are confined to a 2DEG. A common material that is used to build
18
semiconducting heterostructures is GaAs and the g-factor of this material is -0.44 [19]. If
the g-factor of a material is below zero then the spin up electrons in the material will tend
to align parallel to an external magnetic field (Fig.1.3.1).
Figure 1.3.1: The spin magnetic moment of the electron orienting itself to an
external magnetic field where the g-factor is below zero.
In the simulation data shown later the g-factor is not a huge concern. The
simulations incorporate the Zeeman Effect more directly, by allowing the user to simply
control the amount of Zeeman Splitting that the user wants in the system. This technique
is used to show the general operation of the devices.
Effect of Resonant Transmission Through Quantum Dots
In order to implement the Zeeman Effect in the simulations we need to know how
an external magnetic field will change the energy levels of a quantum dot. In these
simulations a quantum dot is modeled as being a simple double potential barrier (Fig.
1.3.2).
19
Figure1.3.2: A simple energy vs. position plot of a double potential barrier system.
This is the basic idea behind the creation of a quantum dot.
A QD will have an intrinsic resonant energy level, in which electrons can exist in
a quasi-bound state. Electrons transmitting through a QD structure will tend to be
transmitted only around the resonant energy of the QD [8]. Using the tight binding
equation, one can build a model of a QD to look at how it transports electrons. In Fig.
1.3.3 is a diagram of a QD with energy qd = 0.
Notice, that when electrons are
transmitted through the device they are indeed moving through the device at an energy
that centers on the resonant energy of the QD (Fig. 1.3.4).
Figure 1.3.3: A quantum dot with one energy level, qd. With the application of the
external magnetic field the energy level splits into an upper and lower state [19].
20
Figure 1.3.4: Plot of T vs. electron E through a single QD. The QD energy is set at 0,
and the energy window is centered on the Fermi energy of the system.
If we apply an external magnetic field, however, to the system then the energy
level will be Zeeman split by some amount z (Eq. 1.3.1) [19]. The electrons that
encounter this system will tend to either align or anti-align with the magnetic field, so
they will either be spin up or down. Remember, however, that the energy of a spin up
electron is lower than that of a spin down. This means that the spin up electrons will tend
to be transmitted through the lower energy level, and the downs through the higher,
assuming a g-factor that is negative (Fig. 1.3.5).
Figure 1.3.5: The resonant transmission through a quantum dot after its energy
level has been spin-split by an external magnetic field.
21
1.4. System Designs
In this thesis we study three systems of QDs; multiple Aharonov-Bohm Rings
(AB Rings) put in series with one another, where intermediate QDs separate the rings,
and an external magnetic field applied to the system in the plane of the rings spin-splits
all the QD energy levels(Fig. 1.4.1, 1.4.2); the multi-AB Ring system, but with spin-split
intermediate QDs (Fig. 1.4.3); a spin-polarizer made of QDs in series with one another,
where one of the dots has had its energy level split with an external magnetic field (Fig.
1.4.4).
In each of the ring systems it is assumed that every ring has a QD in each of its
arms, and that these QDs have been created in a layer of bulk GaAs (Fig 1.4.1). This
material has a g-factor of -0.44 [20], which means that the application of an external
magnetic field produces a measurable change in the Zeeman Splitting Energy (ZSE) of
the QDs. The nanowires comprising the input and outputs leads are also made of a
periodic lattice of QDs, but they have been created in a layer of material that has a gfactor close to zero, so that they are approximately spin-neutral. A good example of a
material that could be used for the leads is InAs [21]. This allows us to spin-split the
energy states in the rings QDs, and examine how this affects the transmission, without
having to worry about the contribution of the leads themselves. The reason for having the
two different materials is that, in practice, the external magnetic field will be difficult to
confine solely to the length of the ring. This makes the possibility of physically creating a
system that reacts as the simulations predict more feasible. Each of the QDs in the ring is
also assumed to have an energy level,
, that can be controlled via Quantum Point
22
Contacts (QPC) acting as voltage gates (Fig. 1.4.1, 1.4.2) There are also QPC which act
to form the QDs and are represented as the potentials between QDs, V (Fig. 1.4.1, 1.4.2).
It is assumed that we can control these as well. In the nanowires leading into and out of
the system the energy of each QD and the coupling between each site, Vo, are not
controllable, but are intrinsic to the material used. Fig. 1.4.2 shows how a physical AB
ring might be constructed, and the QPC used to control the device.
Figure 1.4.1: A single AB Ring with input and output nanoleads and an external
magnetic field parallel to the plane of the ring.
Figure 1.4.2: Construction of a physical AB Ring with quantum dots embedded in
each arm of the ring [22].
23
In the multi-ring systems (Fig. 1.4.3, 1.4.4) we build the QDs in the same way as
the single ring structure, except that we now have rings in series with one another. In the
first case that we treat the intermediate dots are not affected by the application of the
external magnetic field. This could be accomplished with the low g-factor material again.
While these results prove interesting, the system itself may prove to quite impractical to
build. Nevertheless, analysis of the system gives answers which explain some of the
workings of the multi-ring system where the intermediate dots are made from the bulk
GaAs, like the ring QDs. The splitting of the intermediate dots also makes it possible to
calculate the differential spin-polarization for the multi-ring system; something not
possible when they are not split. An important note about the multi-ring system with the
spin-split intermediate dots: when all the QDs in the system have been spin-split, only the
upper levels are connected to each other, and likewise for the lower levels, when the
Tight-Binding Approximation is applied to the system. This means that an electron that is
spin up will not actually have the chance to interact with a spin down electron until it
either reaches the output lead or is reflected back to the input lead.
Figure 1.4.3: Multiple AB Rings in Series with one another. Notice in this diagram
that the intermediate dot is not spin-split.
24
Figure 1.4.4: Multiple AB Rings in Series, where the QDs in the arms of the rings
and the IQDs are all spin-split.
The final system analyzed is a series of quantum dots, where the first quantum dot
is again made out of GaAs, so that its energy level can be spin-split, and the following
ones are made out of something with a g-factor close to zero (Fig. 1.4.5). Two cases are
studied; one where following the spin-split QD there is one QD with an energy level that
can be tuned,
qd,
which then leads to the output of the system; a second where there are
two tunable QDs after the spin-split one, and then the output lead. As before, the input
and output leads are made of a material with a g-factor close to zero. The potentials
between the controllable QDs are all set to the same value, V, also under our control (Fig.
1.4.5). Fig. 1.4.6 shows how a device comprised of QDs in series might be physically
constructed.
Figure 1.4.5: Schematic of the two QD spin polarizer.
25
Figure 1.4.6: Physical construction of quantum dots in series with one another. The
leads forming the QDs are QPC formed with lithography techniques [25].
Chapter 2: Methods of Analysis
2.1.
Energy Scales
In order to link the simulations to experimental work the parameters of the system
are set based on typical voltages, currents, etc. found in the literature. To start, it is
assumed that all of the quantum dots are built in bulk GaAs, which has a g-factor of -0.44
[20]. It is also assumed that the input and output leads of the system are built out of a
substance that has a g-factor relatively close to zero, so that they are spin-neutral and
don’t effect the overall spin-transmission [24,25]. One such material that can accomplish
this is InAs [21]. In general we ignore the effects of the AlGaAs/InAs junctions, but
further work may need to be done in order to take these effects into account. We will also
say that the typical quantum dot energies in the system are around 100 μeV [19]. From
the dispersion relation of the nanowires we know that the energy window of the system is
-2 μeV < E < 2 μeV (Sec. 2.2). From this we can normalize the energies in the plots to the
potentials in the nanowires, Vo, which is set to 1 μeV. Finally, we can now define the
Zeeman energy scales. From (Eq. 1.3.6) we can calculate that:
)B
(2.1.1)
27
This allows us to also calculate the relative magnetic field needed to produce the results
in the simulations. Using the dispersion relation:
.157 T.
(2.1.2)
This means that whenever we want to use the maximum amount of Zeeman spitting
energy available we must use a magnetic field with strength 0.157 T. If the magnetic field
exceeds this value the simulations cease to be realistic.
2.2.
The Tight-Binding Approximation
The Tight-Binding Approximation is a single electron approximation to the
Schrödinger Equation. By applying it to a 2D QD system we can calculate the electron
transmission and reflection of the system as a whole. The general form of this time
independent approximation is:
∑
,
(2.2.1)
where Vn,m is the overlap integral of the potential at site n with the wave at site m, and is
referred to as the coupling of that site ; Ψn is the electron wave function at site n, and
correspondingly for site m; and
n
is the energy of the QD at site n [13, 26]. When
applying this approximation it is important to choose the number of neighbors to include
for each equation, since each QD will get a different equation from the others. In the
simulations, we chose to use the nearest neighbor approximation, which means that when
trying to figure out how the wavefunction at a particular site overlaps with those at other
28
sites the approximation only looks to the wavefunctions at the closest sites. In general it
is possible to include second nearest, third nearest, etc., but for this work the nearest
neighbor approximation should be sufficient, since it is assumed that the individual sites
are far enough apart that only the nearest neighbor contribution is great enough to
consider.
In principle, the Tight-Binding Approximation is an application of Bloch’s
Theorem from quantum mechanics [13]. A line of quantum dots, all with the same
confining potentials and site energies, will constitute a linear, periodic lattice. The
potential of this structure is referred to as the “Dirac Comb” (Fig. 2.2.1).
Figure 2.2.1: Plot of a periodic potential created by a lattice of atoms. Referred to as
the "Dirac Comb" [13].
When trying to solve the Schrödinger Equation for this system we can take the solution to
satisfy the condition:
=
where
potential, and
,
(2.2.2)
, A is a periodic Bloch function that has the same periodicity as the
describes the electron as a plane wave moving through the system. By
using Eq. 2.2.2 as the form of our wavefunction we can build a system of equations using
29
the Tight-Binding Approximation. Applying this to the nanowires that are acting as the
source and drain in the systems we can calculate the dispersion relation [27] for the leads:
(2.2.3)
This relation is quite important, as it allows us to simplify any expression that involves
the input or output leads, and it indicates that the allowed spread of electron energy levels
is
.
Finally, in order to incorporate the Zeeman Effect into the simulation, it is
necessary to treat the Zeeman-split energy levels as two separate energy levels in the
Tight Binding Approximation (Fig. 1.4.1). We generally refer to these as
and
.
This means that each individual QD will actually have two separate equations describing
the electron transport though the QD, one for each energy level. Correspondingly, each
one of these levels will also have its own associated wavefunction in the Tight-Binding
equations. This is what allows for the spin-polarization of the electrons as they traverse
the device, when an external magnetic field is applied. It is important to note that this
does change the simulation slightly, since each QD in the ring structure will now be
coupled to the sites on either side of the ring twice. This actually increases the coupling
to the adjacent sites, in relation to a ring that has no Zeeman Splitting included in it. This
is accounted for by multiplying the couplings in the Zeeman split ring by √ . This is only
necessary, however, if you wish to compare the output of two systems where one
incorporates the Zeeman Effect, and the other does not.
Finally, once we have the system of equations we can solve it quite readily, since
the system can be represented in matrix form:
30
⃗⃗
,
where ⃗⃗ is the matrix formed by the equations,
with ⃗⃗ , and
(2.2.4)
is the vector of constants associated
is the solution vector, which holds all of the unknowns in the system of
equations. Via Mathematica we can invert ⃗⃗ and dot
. Once this is done we simply select the part of
into it, which allow us to solve for
which represents the wavefunction of
the site we are interested in. Taking the absolute value of that wavefunction squared
allows us to then plot that wavefunction as a function of the electron energy. Typically
we plot the
| | and
| | , since the transmission tells me what output to expect
from one of the novel devices [13]. In this case t and r are the coefficients of the
wavefunctions at the input and output of the system. The reflection is useful since
plotting the sum of the transmission and reflection tells me whether or not the simulation
is functioning correctly, since that quantity should always be unity.
2.3.
Weighted Spin-Polarization
The weighted spin-polarization of the transmission of one of the devices is a
mathematical technique whereby the degree to which the transmission has been spinpolarized can be readily represented. It is useful to represent the transmission in this way,
because it allows us to find useful regions of parameter space; those that switch the spinpolarization with only a small change in the parameters of the system; those which create
a maximal spin-polarized transmission, while reducing the other significantly; those
which create a spin-polarized transmission that, experimentally, would be easy to
31
measure or manipulate. In order to calculate the spin-polarization we first calculate the
transmission of a system through only the spin up states, and then through the spin down
states, i.e. we must calculate the transmission twice. Then, we can find the spinpolarization term, |
|, which is useful because even if the transmissions are
small, the ratio may still approach unity. This helps to extract the degree of spinpolarization, even when the transmission peaks are quite narrow. When the spinpolarization term is multiplied by the transmission of one of the states, we find the degree
of spin-polarization:
|
|
|
|
(2.3.1)
.
(2.3.2)
The polarizations of the energy state, once calculated, can also be used to find the
differential spin-polarization of the system:
.
(2.3.3)
This tells us the change in the weighted spin-polarization, and also makes finding the
degree of spin-polarization produced much easier to visualize, since all spin up states will
have positive values, and spin down states will have negative values, or vice versa
depending on the sign of the g-factor of the material used.
32
2.4.
Calculation of IV Curves
In order to calculate the overall theoretical current that is passing through one of
the devices, the electron transmission of the system must first be found. Each system has
input and output leads which correspond to the source and drain. In an experimental setup these two contacts will be kept at different voltages, which will force the source and
drain to have different electrochemical potentials. The applied voltage bias across the
system can be defined as:
(2.4.1)
where V is the applied voltage, q is the electron charge, and μ1 and μ2 are the
electrochemical potentials of the source and drain respectively [28]. The source and drain
will locally be in equilibrium, since we are providing a constant voltage source. This
means that both the source and drain will have their own, constant, Fermi functions:
,
(2.4.2)
,
(2.4.3)
where E is the energy of an electron in the system, kB is Boltzmann’s constant, and T is
the temperature of the system [28]. The source would like the number of electrons at its
Fermi energy to be
, while the drain would prefer
This means that the drain
will try to reduce the Fermi energy of the source by pulling electrons away from it and
then sending them back to the power supply. The source, however, will replace those
electrons with those coming from the source. In this way a steady state number of
33
electrons, N, somewhere between
and
will arise in the system, as well as a
steady current. Using the Fermi functions of the source and drain, as well as the electron
transmission through the system we can now calculate the current of the overall device:
∫
,
(2.4.3)
where h is Planck’s constant [28]. This calculation is generally correct; however it is
important to remember that we are assuming, by using this formula, that the current is a
coherent stream of electrons. This means that none of the electrons undergo phasebreaking scattering processes that cause a change of state in an external object. More
succinctly, this means that when an electron scatters off part of the lattice that it is
confined in we assume that the lattice does not vibrate, or at least not very much.
Chapter 3: The Aharonov-Bohm Ring
3.1. Simulation Parameters
In the simulations of this chapter the site energies for the nanoleads,
0.0, as well as the site energies of all the intermediate QDs,
o
o,
are set to
(Fig. 3.1.1). The coupling
parameters between sites in the nanoleads, Vo , are set to 1, and this value is used as a unit
of energy through the discussion. Each one of the rings has an asymmetric configuration,
meaning that the site energy of the lower dot is equal in magnitude to that of the other,
but always opposite in sign (i.e. odd QDs have Ɛ = 0.1 and even have Ɛ = -0.1) . Any
coupling constant leading to a QD in the system is set to 0.1. The external magnetic field
being applied to the system is parallel to the plane of the ring structure, so that no flux is
present. The limit to the external magnetic field is 0.157 T, which translates to a Zeeman
Splitting energy of ez = ± 2 μeV. This keeps the system within the energy window as
prescribed by the dispersion relationship (Sec. 2.1). The number of rings in the system,
the electron energy, and the amount of ZSE are the varied parameters.
35
Figure 3.1.1: Schematic of a double AB Ring with an applied external magnetic
field.
3.2. The Multi-AB Ring System
The single AB ring (Fig. 1.4.1) is a well-studied system, in which the Zeeman
Effect and the Aharanov-Bohm Effect have been used to characterize the system in great
detail [26]. Further work with this system has also shown that, under certain conditions,
the AB ring can spin-polarize an electron transmitted through the system. In this work we
endeavor to further characterize the AB ring by linking a multitude of them in series with
one another and by utilizing the Zeeman Effect to spin-split the energy levels in the
systems QDs. The main objective in doing this is to generate a system which can spinpolarize a current efficiently. The results show that a series of these rings does not
immediately lend itself well to being used as a spin-polarizer, but may be of use as a
single electron transistor (SET), a nanoelectronic device [29,30].
A single QD will transmit electrons that are at or near the quasi-bound energy
state of the QD (Fig. 1.3.4). The peak in Fig. 1.3.2 can be moved along the x-axis of the
36
system by simply changing the energy of its resonant energy level, which would
represent higher energy electrons transmitting through the QD if you increase the QD site
energy. The width of the peak can also be manipulated by increasing the couplings
around the QD, which increases the width of the peak, and vice versa. In this way we can
directly control the output transmission of the system. By applying an external magnetic
field we can spin-split the energy level of the QD, and in this case resonance peaks will
form around these two new energy levels (Fig. 1.3.5). These basic techniques for
manipulating the transmission of a QD apply to a system of QDs as well (Sec 1.3).
In the case of the asymmetric AB ring we have two QDs, each with an
independent energy level, through which the electrons can travel. As such the
transmission will look quite similar to that of the single Zeeman split QD, but the current
produced by the system will have no spin-polarized elements (Fig. 3.2.1). Applying an
external magnetic field spin-splits both of the AB ring’s QDs, giving the electrons four
potential paths to traverse (Fig. 3.2.2). This, however, means that the energy levels
interlace with each other, requiring a greater magnetic field to separate the spin-states
energetically.
37
Figure 3.2.1: Transmission vs. electron energy through a single AB ring without an
applied external magnetic field. Electron transmission is occurring at the quasibound energy state of each of the QDs, where ƐQD = ±0.1.
Figure 3.2.2: Transmission vs. electron energy through a single AB ring with
Zeeman splitting energy of 0.1.
The interesting phenomenon of this system arises with the addition of rings to the
system. In Fig. 3.2.3 is a set of transmission results for the multi-ring system, where the
intermediate dots are not spin-split (Fig. 1.4.3).
38
Figure 3.2.3: Transmission through the unsplit intermediate dot multi-ring system
for two(a-c), five (d-f), ten (g-i), and twenty (j-l) ring structures. The leftmost graphs
show no ZSE. The central graphs are for a ZSE of 0.05, and the right most 0.1.
With every additional ring added to the first an additional pair of peaks is added
to the transmission. These extra resonances always have a higher magnitude energy than
the peaks corresponding to the quasi-bound energy state that the QDs have been set at. As
the ring number increases the number of these resonant peaks increases, but the energy
range in which they appear stays constant, -0.4 < E < 0.0 for the peaks corresponding to
the lower QDs’ energy levels, and 0.0 < E < 0.4 for the upper QDs’. Notice that the edges
39
of the band gaps are at a slightly higher energy than that of the QDs. This effect is
magnified as the site energies are set farther away from 0, the Fermi energy of the
system. At high ring number, the peaks narrow and begin to form a continuum. Two
conduction bands, separated by a band gap that is approximately 2
qd,
form, as rings are
added, so the system resembles a semiconductor (Fig. 3.2.3). By adjusting the QD site
energies in the system the conduction bands can be manipulated, creating a smaller or
larger band gap. This is done in real life by adjusting a gate voltage that is not used to
confine the electrons (Fig. 1.1.4). This is quite interesting, since the band gap of a typical
semiconductor cannot be controlled, but is an intrinsic property of the material.
Unfortunately for this particular set up, the application of an external magnetic field
produces rather strange results. The outer peaks, formed by the addition of extra rings
beyond the first, are not spin-split by the magnetic field. This is unrealistic, since any
electron energy state in a magnetic field will split into two states. The reason this occurs
is the physical construction of AB rings would prevent us from allowing the intermediate
QD between the rings to remain unaffected by Zeeman splitting. This manifests itself in
the program first, by not spin-splitting the outer peaks, but also in the strange form of the
resonances created when the external field is added (Fig. 3.2.4, 3.2.5).
40
Figure 3.2.4: Plot of transmission vs. Zeeman splitting energy and electron energy
for the multi-ring system. Notice that the outer peaks do not split like the central
ones.
Figure 3.2.5: Further transmission plots for the multi-ring system, showing the
transmission at ZSE of 0.2, 0.3, and 0.4, from left to right (Fig. 3.2.4).
41
Even though the simulation may not be very realistic, the data produced are still
very instructive when trying to understand the outer peaks in the transmission at zero
external magnetic field. First, the outer peaks are always present at energies higher in
magnitude than that of the QD site energy associated with the conduction band in
question. This is also true when the central peaks have been spin-split; the outer peaks
increase in energy in order to remain at an energy that is higher than its peak in
magnitude (Fig. 3.2.4, 3.2.5).
In order to produce more realistic results, the intermediate QDs are now spin-split
in the simulation. From Fig. 3.2.6 it is evident that as we apply an external magnetic field
to the system all of the resonant peaks become spin-split, which is what we would expect
(Fig. 3.2.3 3.2.6). This is good, since the fact that all the peaks become spin-split only
when the intermediate quantum dots (IQD) are as well demonstrates that in the previous
simulation the transmission was limited by the unchanging IQD site energy. In effect,
that site energy not being affected by the magnetic field demonstrates a few things when
compared to the spin-split IQDs; the outer peaks are indeed caused by the addition of the
rings; the transmission through multiple rings is highly dependent on the IQD; spinpolarized transmission through unsplit IQDs becomes “bottlenecked” at the unsplit IQD,
effectively creating a system of individual rings that transmit electrons to each other,
rather than a system comprised of many rings in series.
42
Figure 3.2.6: Transmission through multi-ring systems for two (a-c), five (d-f), then
(g-i), and twenty (j-l) ring structures. The leftmost graphs show no applied external
magnetic field. The central graphs show a ZSE of 0.05, and the right most 0.1.
43
To further understand the multi-ring system, the external magnetic field is applied
in order to study the effect it has on the transmission (Fig. 1.4.4). At 0.0 Zeeman splitting
energy (ZSE) the transmission is identical in form to that of the unsplit IQD system. The
band gap is still slightly wider than 2
the energy ranges -0.3Vo < E < -
qd
and
qd,
and the conduction bands are now confined to
qd <
E < 0.3Vo. Like the unsplit IQD simulations,
adding rings to the system causes the conduction bands to become better defined, as the
transmission peaks become sharper and closer together. The application of an external
magnetic field now splits all of these resonant peaks, causing the conduction bands to
become even more of a continuum. Increasing the ZSE further causes the spin-polarized
bands to start crossing over one another, reducing the band gap to zero when the ZSE is
on the order of the QD energy levels (Fig. 3.2.6). Increasing the ZSE past three times that
of the QD site energies causes the spin-polarized conduction bands to separate again,
forming three individual band gaps in the transmission (Fig. 3.2.7).
44
Figure 3.2.7: Plot of the five ring system with varied ZSE.
Figure 3.2.8: Transmission results for the 5 ring system with spin-split intermediate
dots. The amount of ZSE is (a) 0.2, (b) 0.3, and (c) 0.4, from left to right (Fig. 3.2.7).
45
One way we can further characterize this system, and to evaluate its potential as a
nanoelectronics/spintronic device, is by calculating the tunneling current through the
system (Sec. 2.4) and plotting the I-V curve of the system. In these calculations we
assume that the leads are symmetric,
⁄ , where Vsd is the applied source-drain
voltage, and that the Fermi energy, EF = 0.0. This source drain voltage will be varied,
allowing us to characterize the tunneling current over the whole energy window of the
transmission T. Since the energies of the QDs are controllable, the band gap can be
manipulated easily. Fig. 3.2.9 contains I-V plots corresponding to the transmission plots
in Fig. 3.2.6.
When no external magnetic field is present the systems I-V characteristics
resemble that of a semiconductor [30] (Fig. 3.2.9 a, d, g, j). Increasing the ZSE to the
order of the QD site energy (
qd
= 0.1) produces a linear I-V curve that resembles a
conductor (Fig. 3.2.9 c, f, i, l). By increasing the ZSE further, band gaps form in the
system for energies
qd
> 0.3 and the system returns to semiconductor I-V characteristics,
but now with multiple band gaps (Fig. 3.2.6). Notice that the system’s I-V characteristics
start to saturate after 5 rings have been included, meaning that past this number the I-V
curves change to a lesser degree with the addition of more rings. For this reason the five
ring structure is studied further, since it can provide a great deal of information about the
system without sacrificing simulation time.
46
Figure 3.2.9: I-V Plots for the multi ring structure. The plots are arranged in similar
fashion to those in Fig. 3.2.6, and each I-V plot is associated with the same
transmission plot. Adding more rings makes the system more distinctly a
semiconductor, until five. With more than five the system saturates and further
change in the conductance is not plainly seen.
47
The relative spin-polarization of the five ring structure is also important to the
discussion, and in Fig. 3.2.10 the contour plot shows how the differential spinpolarization (Sec. 2.3) of the multi-ring system evolves, with added ZSE. Fig. 3.2.11
contains sample differential spin-polarization plots to better illustrate the spinpolarization of the system. A majority of the time, with the application of the ZSE, the
system is in a spin-mixed state. In other words, the spin-polarized transmission peaks
have not been differentiated from one another in energy. At ZSE = 0.4 the system
becomes fully spin-polarized, as both of the spin-states have been completely
energetically separated from one another. The system presents an interesting quality then,
since in previous work [26] the single AB ring was quite capable of producing similar
spin-polarization, but only with a few resonant peaks for each state. Experimentally this
would be difficult to measure, since the narrow resonant transmission peaks only allow a
few electrons with a small energy window through the system. In this case, the
conduction bands allow for a broader energy spectrum of electrons to be transmitted,
meaning the spin-polarized currents produced will be stronger, and therefore easier to
measure. In all the plots shown, the inter-dot coupling integrals were set to Vn = 0.1. If we
increase these couplings, the resonant peaks in these conduction bands will broaden,
generating a larger area of spin-polarization in the system (Fig. 3.2.12, 3.2.13, 3.2.14)
[26]. In this way, the multi-ring system is acting as an even better spin-polarizer. In both
cases, however, the spin-polarized currents are not physically separated at the output of
the device, so a way to isolate the currents needs to be found.
48
Figure 3.2.10: Plot of the differential spin-polarization for the 5 ring system with
varied amount of ZSE.
Figure 3.2.11: Plots of the 5 ring structure’s ΔPol with changing amounts of ZSE.
The first plot, a, shows EZ=0.0, b EZ=0.1, c EZ=0.2, d EZ=0.3, e EZ=0.4, and f
EZ=0.5 (Fig 3.2.10).
49
Figure 3.2.12: The transmission of the five ring system with varied ZSE. In this case
the coupling has been increased from V = 0.1 to V = 0.3.
Figure3.2.13: Transmission plots accompanying the contour plot above. The first
plot shows (a) EZ = 0.0, and this value increases with each graph by 0.1, until (e),
which is EZ = 0.4 (Fig. 3.2.12).
50
Figure 3.2.14: Contour plot showing the differential spin-polarization vs. EZ and E.
This is the relative spin-polarization for the system in Fig. 3.2.13. Notice that wide,
spin-polarized conduction bands have been generated by the system.
3.3. Applications
From the I-V characteristics, we propose that the multi-ring system may be useful
as a nanoelectronic device, the single electron transistor (SET). The typical SET operates
using a gate voltage to control the tunneling current through the Coulomb Island in the
device [29, 30]. The multi-ring system may serve the same function, except the external
magnetic field manipulates the overall I-V characteristics. An incident electron, close to
51
the Fermi energy of the system, would effectively be blocked by the free standing
semiconductor configuration, where ZSE = 0.0, but be allowed to pass if the system is
switched to a conductor in that energy range.
In the realm of spintronics, this device may prove quite useful as either a spintransistor or spin-polarizer. The ability of the system to create a wide energy range spinpolarized current will potentially give experimentalists the ability to generate a spinpolarized current large enough, and refined enough, to be useful as a power supply. The
problem remains, however, that the system is not able to physically separate the two
currents.
The device may also be useful as a spin-transistor. We have already seen its
potential as a SET, but if the system is made able to physically differentiate between the
spin-states, by effectively filtering the other out, then it could produce a current that uses
the spin-states as signal, switching between transmitting one state or the other. Notice,
however, that we need a way to physically separate the spin-polarized currents in order
for this, or the spin-polarizer to work. This issue is addressed in a simple model of the
spin-polarizer.
Chapter 4: The Quantum Dot Spin-Polarizer
4.1. Simulation Parameters for the Quantum Dot Spin-Polarizer
In the simulations of this chapter, the site energies for the nanoleads are set to
o=
0.0. The coupling parameters between sites in the nanoleads, Vo, are set to 1, and this
value is used as a unit of energy through the discussion (Fig. 4.1.1). The site energies for
the two quantum dots will be equal, unless otherwise noted. Only the first QD will be
affected by the Zeeman Effect, and it will have a site energy of
qd
= 1.0. The inter-dot
coupling integrals of any path leading to a QD are set to V = 0.2. The limit to the external
magnetic field is 0.157 T, which translates to a Zeeman Splitting energy of EZ = ± 2 μeV.
This keeps the system within the energy window as prescribed by the dispersion
relationship (Sec. 1.3). The couplings between dots and the energy levels of the dots, and
the ZSE are the varied parameters.
53
Figure 4.1.1: Schematic of the double quantum dot spin-polarizer with applied
external magnetic field.
4.2. The Double Quantum Dot System
The difficulty of physically separating the spin-states of a current which has
energetically separated spin-polarized currents is now addressed. To start, the system’s
reaction to an external field when the filtering quantum dot (FQD) has a site energy of
zero is exactly that of the single Zeeman split QD attached to input and output nanoleads
(Sec. 1.3). When the FQD’s site energy is increased, the Zeeman split energy level that is
closest in energy to the FQD tends to be favored over the other (Fig. 4.2.1). When the
energy of the FQD equals that of one of the Zeeman split levels, that level will transmit at
a significantly higher energy than the other, but only if the ZSE is high (Fig. 4.2.1, 4.2.2).
If we increase the FQD site energy past that of the favored Zeeman split energy level then
its transmission begins to be filtered as well, but not to the extent of the other, which will
be reduced even further (Fig. 4.2.1). To optimize the spin-polarization of the system we
add enough ZSE to push the resonant peaks to the edges of the energy window (Fig.
4.2.3). Tuning the FQD to one of the energy levels we can achieve a moderately spinpolarized current, where the electrons with energy equal to the favored site energy are
transmitted at 100%, and those of the other peak only a little under 40% (Fig. 4.2.3).
54
Figure 4.2.1: Transmission plots for the DQDS. In each plot the vertical red line is at
the site energy of the FQD. The two vertical black lines are at the site energies of the
Zeeman split energy levels.
55
Figure 4.2.2: Transmission plot from the table above, where the site energy of the
FQD equals that of the higher energy Zeeman split state.
Figure 4.2.3: The basic optimized DQDS. The amount of Zeeman splitting energy is
increased to 1.9, and the FQD’s energy level is changed to match the upper peak.
Notice that the transmission of the peak furthest from that of the site energy of the
FQD is reduced by a little over 60%, a good start to producing a spin-polarized
current.
56
4.3. The Triple Quantum Dot System
The triple quantum dot system (TQDS) produces the same results as the double
quantum dot system (DQDS) when the site energy of the two FQDs are centered in the
energy window (Fig. 4.3.1).
Figure 4.3.1: TQDS transmission plot that corresponds to Fig. 4.1.1 for the DQDS
(Fig. 4.2.1). The ZSE is 1.0 again.
In the identical case to that above, where the ZSE pushes the resonant peaks of the
spin-split QD to the edge of the energy window, and the site energies of the FQDs match
that of one of the spin-split energy states, the transmission of the favored resonant peak is
at 100%. The transmission of the other peak, however, is reduced further when compared
to the DQDS, so that it is nearly non-existent (Fig. 4.3.2, 4.3.3). Adding a second FQD,
then, has produced an even more efficient spin-polarizer. Manipulating the couplings in
the TQDS allows us to create transmission peaks which cover a wide energy range (Fig.
4.3.4, 4.3.5, 4.3.6). This provides an advantage, experimentally, since the wider energy
57
band will allow more electrons to tunnel through the system, creating a larger, more
readily measured tunneling current. In this case there are multiple optimizations.
Figure 4.3.2: A table of transmission plots for the TQDS which corresponds to Fig.
4.2.1, so that the DQDS and TQDS can be compared.
58
Figure 4.3.3: The TQDS optimized, with the same parameters as those of the DQDS
in Fig. 4.2.3 (ZSE = 1.9, V = 0.1, ƐQD = 1.9).
Figure 4.3.4: The optimization of the TQDS transmission, where the coupling
between sites is V = 1.3, the ZSE = 1.9, and the site energy of the FQD is FQD = 1.9.
59
Figure 4.3.5: Contour Plot of the TQDSP, where EZ = 1.9, the site energy of the
FQD is FQD = 1.9, and the interdot coupling V is varied to show how the width of
the spin-polarized transmission peak changes.
Figure 4.3.6: Plots showing the ΔPol of the transmission in Fig. 4.3.5 a) The
differential spin-polarization of the TQDS when V = 1.3, b) V = 0.55.
60
The difficulty with this system is that, while the transmission is covering a wide
range of energies, the spin-polarization of the transmission is not immediately obvious.
Notice in Fig. 3.4.6 (a) the spin-polarization at V = 1.3 is actually spin-mixed. In Fig.
3.4.6 (b) V = 0.55, and the system is spin-polarized. The disadvantage, then, is the
transmission of spin-polarized electrons is only around 65% at maximum, but it still
covers a wide energy range, making it feasible that the TQDS could be used to produce a
spin-polarized tunneling current.
4.4. Applications of the Quantum Dot Spin-Polarizer
Both the DQDS and TQDS are relatively simple models, and both seem to spinpolarize the transmitted electrons quite well, with the TQDS achieving approximately
100% in some cases. While this simple device may potentially serve as a spin-polarizer
quite well, it also demonstrates that the FQDs are capable of excluding one spin-state
from being transmitted, while allowing the other to pass. This result is quite important;
since in Sec. 3.3 it was noted that without a method for excluding one spin-state from the
transmission, the device would only spin-polarize the current energetically. To this end,
adding FQDs to the multi-ring system may prove very beneficial, in that it could become
a true spin-polarizer.
Chapter 5: Future Work
The next step to be taken in this research, first, is to combine the multi-ring
system with the FQD concept, so that simulation results can show if the system can be
turned into a spin-polarizer that creates a wide spin-polarized conduction band.
In order to further characterize the AB ring, it is also important to note that no
mention of the AB Effect is made when analyzing the multi-ring system. Additional
programming is necessary to make it capable of calculating the electron transmission if
magnetic flux penetrates each of the rings (Append. A). It will also be important to study
how the AB Effect and the Zeeman Effect alter the multi-ring results when coupled
together. It will also be important for us to further manipulate the site energies and
couplings in the system.
Finally, at some point it will also be necessary to physically build the devices
devised. Without experimental work the system can never be proven in real life. This is
necessary if any of the units are to be made practical computing components.
Chapter 6: Summary
In this research we apply the Tight Binding Approximation to quantum dot
systems and study the electron transmission produced at the output. The overarching goal
of the work was to try to create a device schematic that could serve as a spin-polarizer, or
as some other useful spintronic or nanoelectronic device. In spintronics the need for an
efficient spin-polarizer is quite large, since without it we cannot generate a current usable
by all other spintronic systems.
Experimentally, many spin-polarizers have been
generated, but their efficiency was too low to be promising as a spintronic power supply
(Sec 1.1).
To this end, simulations using the Tight-Binding Approximation were used to
model two systems, the Multi-Aharonov-Bohm Ring and the Quantum Dot SpinPolarizer (Sec 1.4, Ch. 2). For the multi-AB ring system, the number of rings in the
system was varied along with the strength of an external magnetic field. We find that the
multi-ring system I-V characteristics are akin to those of a semiconductor, because of the
band gap present in the transmission. With the application of an external magnetic field,
the energy states of the system are spin-polarized, producing more resonant peaks and
making the semiconductor qualities even better defined, when the magnetic field is small.
As the field strength is increased, the band gap decreases and the system eventually
63
becomes a conductor. Further increasing the field strength causes the system to become a
semiconductor again, but with three band gaps instead of the original one. At this point,
when the system has become a semiconductor again, the spin-polarized states of the
system have also been energetically isolated from one other, forming wide spin-polarized
conduction bands. The switching from semiconductor to conductor makes the device
potentially useful as either a single electron transistor, a nanoelectronic device, or a spintransistor. The ability to produce wide spin-polarized conduction bands makes it a good
candidate for a spin-polarizer as well. A problem arises, however, in that the system does
not physically separate the spin-polarized currents it produces. This would be necessary if
it were to function as either spin-polarizer or spin-transistor.
The quantum dot spin-polarizer is proposed first as a solution to the problem of
producing spin-polarized currents for spintronic devices. It also is possible that the results
from this device may be applicable to the multi-ring system, so that it can function as a
spin-polarizer/transistor. Results are promising, in that the DQDS is able to reduce the
transmission probability of one of the spin-states by 60%, while leaving the other
relatively untouched. The TQDS goes even further, and reduces the other spin-state to
almost 0% transmission, while allowing the other to pass, effectively generating a spinpolarized tunneling current. Manipulation of the couplings in the TQDS also provides
interesting results, where increasing the coupling creates a spin-polarized conduction
band that covers a large portion of the energy window. This is advantageous, since,
should the device be built, the tunneling current would be large enough to be readily
measured experimentally.
64
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66
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67
Appendix A:
Master Program for the Multi-Ring System
Clear["Global`*"]
(*This program will generate a matrix of equations based on the tight binding
approximation. The system being modeled is a series of Aharonov Bohm rings.*)
(*This part of the program defines a matrix of zeros, sized for the number of
equations we want to solve. It defines a new matrix, square, sized for the
number of quantum dot energies. The second matrix is parsed into the first.
This gives the diagonal term of energies. After that a loop creates an array
with all the QD energy levels (generates variables). After that it assigns a
value to the upper dots(odd numbered) and to the lower dots(even numbered)*)
(*The variables defined allow the user to simply pick the number of rings in
the system.*)
numring=n = 2;
numidot = m=n-1;
numlead=2;
numrdot = 4*n;
(*This number accounts for the number of zeeman split energy levels, not the
original number before the addition of the magnetic field.*)
matdimension = d1=numidot+numrdot+numlead;
submatdimension =d2 = d1-2;
(*"main" is the primary matrix that we’re trying to build up. The matrix will
be all zeroes and is the exact size needed to solve the system of equations
generated for the multi-ring system.*)
main = Array[0&,{d1,d1}];
MatrixForm[main];
(*Generates a submatrix that is smaller in both dimensions than the main one by
2. The energy terms for all of the wavefuncionts form a diagonal that does not
sit on the main diagonal. I use "around" as an intermediate sub matrix that
will be inserted into "main"*)
sub = Array[0&,{d2,d2}];
MatrixForm[sub];
(*Generates an array that holds all of the values for the onsite energies.*)
energy = Array[#&,{d2}];
MatrixForm[energy];
(*The loop goes through around and assignes values along it's diagonal. Note
that I'm also including the fermi energy of the system, as prescribed by TBT.*)
Do[sub[[i,i]]=e-energy[[i]],{i,d2}];
MatrixForm[main];
MatrixForm[sub];
68
(*Inserting around into the appropriate place inside of main, without
disturbing any other part of main. It's important to do this part first. If you
use this technique and start with other things, like coupling constants, you
will inadvertently delete some things when you insert the energies.*)
main[[2;;d1-1,3;;d1]] = sub;
MatrixForm[main];
(*Everything above was symbolic. The two loops show how I could assign energies
to, say, all the top and bottom quantum dots. You don't have to do it this way,
but I want a periodic system, so I'll assume that the energies of the sites
are, in some way, uniform.*)
(*Typical Bloodshed would define the energies here, but the differential spin
polarization needs them defined in the last section of this program in the spin
up and down modules. *)
Do[energy[[i]]=uu,{i,1,d2-m,4}];
Do[energy[[i]]=ul,{i,2,d2-m,4}];
Do[energy[[i]]=lu,{i,3,d2-m,4}];
Do[energy[[i]]=ll,{i,4,d2-m,4}];
(*The next line defines the energy level of the intermediate dots when they
aren't being spin split. When they are I refer to a variable called "em"*)
Do[energy[[i]]=eem,{i,d2-m+1,d2,1}];
MatrixForm[energy];
(*Insterting quantum dot energy values into the main matrix.*)
Do[sub[[i,i]]=e-energy[[i]],{i,d2}];
main[[2;;d1-1,3;;d1]] = sub;
MatrixForm[main];
(*Generate an array to hold all the values for the coupling constants. Assuming
that these are also periodic, like the on site enregies.*)
couple = Array[V#&,{4*n+1},0];
MatrixForm[couple];
couple[[1]]=1.0;
Do[couple[[i]]=0.1,{i,2,4*n+1}];
MatrixForm[couple];
(*Generate another array that will hold the arrays of constants that will also
be inserted into "main".*)
parts = Array[0&,{n-1}];
MatrixForm[parts];
(*The coupling constants are inserted into arrays in a periodic way.
matrix for two or three rings to see what this means.*)
Do[parts[[i]]={{couple[[4*i]],couple[[4*i]],
couple[[4*i+1]], couple[[4*i+1]],
couple[[4*i+2]], couple[[4*i+2]],
couple[[4*i+3]], couple[[4*i+3]]}},
{i,n-1}];
Look at a
69
(*The arrays of coupling constants are inserted into "main" as well as their
transposes. The indecies for the insertion are easy to figure out, you just
have to think about where the arrays need to be inserted (again, check out a
tbt matrix for multiple rings. there is a pattern).*)
Do[main[[(numrdot+i+1);;(numrdot+i+1),3+4*(i-1);;10+4*(i1)]]=parts[[i,1]],{i,n-1}];
Do[main[[(2+4*(i-1));;(9+4*(i1)),(numrdot+i+2);;(numrdot+i+2)]]=Transpose[parts[[i]]],{i,n-1}];
(*The equation for n=0 in my model always has the same form and number of
elements, regardless of the size of the system. I'm going to manually write the
arrays and simply insert them in the appropriate spots in "main".*)
n0
I*],0,couple[[2]],couple[[2]],couple[[3]],couple[[3]]}};
MatrixForm[n0];
={{-couple[[1]]*Exp[-
main[[1;;1,1;;6]]=n0;
nr ={{couple[[2]]},{couple[[2]]},{couple[[3]]},{couple[[3]]}};
MatrixForm[nr];
main[[2;;5,1;;1]]=nr;
MatrixForm[main];
(*This next array builder will generate the equation for n=1, in my numbering
convention. Make sure to look at the convention I'm using to organize my
matrix.*)
n1 = Array[0&,{1,d1}];
MatrixForm[n1];
n1[[1,2]]=-couple[[1]];
MatrixForm[n1];
(*I'm Using n11 as an intermediate array to help construct n1. n12 will be used
to construct the t column of the matrix.*)
n11 ={couple[[(4+2n)]],couple[[(4+2n)]],couple[[(5+2n)]],couple[[(5+2n)]]};
n12
=Exp[I*]*{{couple[[(4+2n)]]},{couple[[(4+2n)]]},{couple[[(5+2n)]]},{couple[[(5
+2n)]]}};
MatrixForm[n11];
n1[[1;;1,3+4*(n-1);;6+4*(n-1)]]=n11;
MatrixForm[n1];
main[[d1;;d1,1;;d1]]=n1;
main[[2+4*(n-1);;5+4*(n-1),2;;2]] = n12;
(*After this step the entire main matrix has been built.*)
MatrixForm[n12];
MatrixForm[main];
70
(*Generating an array that will hold the basic values needed to create the
constants used in the matrix equation. In any system of rings the constants
will always start with these five. After that, you just need the appropriate
number of zeros to fill in the rest. This is easily done, since we already have
a value for the main matrix dimensions. *)
constants = Array[0&,{d1,1}];
beginconstants={{couple[[1]]*Exp[I*]},{-couple[[2]]},{-couple[[2]]},{couple[[3]]},
{-couple[[3]]}};
MatrixForm[beginconstants];
constants[[1;;5]] = beginconstants;
MatrixForm[constants];
MatrixForm[main];
(*At this point the matrix and constant vector are complete,
intermediate dots having been spin split.*)
(*___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
___ ___ ___ ___ ___ *)
(*This part of the program will
intermediate dots are spin split.*)
rework
the
matrix
but without the
___
___
___
___
"main",
___
___
___
___
so
___
___
___
___
that
___
___
___
___
the
(*This part of the program will recalculate the size of the matrix. Spin
splitting the intermediate dots means you need to add some extra rows and
columns.*)
em=0;
gutdr=d1-m-1;
gutdc=d1-m;
diffd=d1+m;
MatrixForm[parts];
(*Define the new matrix that is "different" from "main", hence the name.*)
maindiff=Array[0&,{diffd,diffd}];
(*MatrixForm[maindiff]*)
(*Gut the old matrix "main", and put the useable bits into the new matrix.*)
maindiff[[1;;gutdr,1;;gutdc]]=main[[1;;gutdr,1;;gutdc]];
maindiff[[diffd;;diffd,1;;gutdc]]=main[[d1;;d1,1;;gutdc]];
MatrixForm[maindiff];
71
(*This defines a small array of coupling constants. In the tight binding there
are zeros between the couplings in the portion of maindiff that characterizes
the intermediate dots.In main there is no separation.*)
Do[parts[[i]]={{couple[[4*i]],0,
couple[[4*i+1]], 0,
couple[[4*i+2]], 0,
couple[[4*i+3]], 0}},
{i,n-1}];
MatrixForm[maindiff];
(*This generates a small tensor that holds two arrays to account for the
splitting apart of the couplings for the intermediate dots. This does the
columns. Subgen2 takes the transpose of these parts, manipulates them a little,
and then does the rows. Once the appropriate arrays are built they are inserted
into maindiff*)
subgen1=Array[0&,{2,9}];
gen1=Array[0&,{m}];
Do[gen1[[i]]=subgen1,{i,1,m}];
MatrixForm[gen1];
MatrixForm[gen1[[1]]];
Do[
gen1[[i]][[1;;1,1;;8]]=parts[[i]];
gen1[[i]][[2;;2,2;;9]]=parts[[i]],
{i,1,m,1}];
MatrixForm[gen1[[1]]];
MatrixForm[gen1];
Do[maindiff[[(gutdr+1)+2*(i-1);;(gutdr+2)+2*(i-1),3+4*(i-1);;11+4*(i1)]]=gen1[[i]],{i,1,m,1}];
MatrixForm[maindiff];
subgen2=Array[0&,{9,2}];
gen2=Array[0&,{m}];
Do[gen2[[i]]=subgen2,{i,1,m}]
MatrixForm[gen2];
MatrixForm[gen2[[1]]];
Do[
gen2[[i]][[1;;8,1;;1]]=Transpose[parts[[i]]];
gen2[[i]][[2;;9,2;;2]]=Transpose[parts[[i]]],
{i,1,m,1}]
MatrixForm[gen2[[1]]];
MatrixForm[gen2];
Do[maindiff[[2+4*(i-1);;10+4*(i-1),(gutdc+1)+2*(i-1);;(gutdc+2)+2*(i1)]]=gen2[[i]],{i,1,m,1}];
MatrixForm[maindiff];
(*This generates the array of spin split energy levels in the intermediate
dot.*)
energyint=Array[0&,{2m}];
MatrixForm[energyint];
Do[energyint[[i]]=e-(em+ez),{i,1,2m,2}];
Do[energyint[[i]]=e-(em-ez),{i,2,2m,2}];
72
(*Inserting the new energies into maindiff.*)
MatrixForm[energyint];
Do[maindiff[[gutdr+i,gutdc+i]]=energyint[[i]],{i,1,2m,1}];
MatrixForm[maindiff];
(*The array of constants will be the same, but will have more zeros. This guts
"constants", builds a new array of appropriate length for maindiff, and then
inserts the constants into the appropriate spot in the array.*)
diffconstants = Array[0&,{diffd,1}];
diffconstants[[1;;5]] = beginconstants;
MatrixForm[diffconstants];
(*___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
___ ___ ___ ___ ___ *)
___
___
___
___
___
___
___
___
___
___
___
___
___
___
___
___
___
___
___
___
(*This part of the program will calculate the differential spin polarization.*)
mainup=Array[0&,{d1,d1}];
maindown=Array[0&,{d1,d1}];
mainup[[1;;d1,1;;d1]]=main[[1;;d1,1;;d1]];
maindown[[1;;d1,1;;d1]]=main[[1;;d1,1;;d1]];
MatrixForm[mainup];
MatrixForm[maindown];
(*Now that there are two matrices, one for spin up and one for spin down, we
must calculate the transmission for each path seperately.*)
mc1:=Inverse[mainup];
mc2:=Inverse[maindown];
JBF1:=mc1.constants;
t1:=Part[JBF1,2];
JBF2:=mc2.constants;
t2:=Part[JBF2,2];
(*Calculation of transmission is typical, but I must define the QD energies in
the modules, otherwise their values will interfere with each other.*)
tup[en_,eez_]:=Module[{},
e:=en;
eu=0.1;
el=-0.1;
ez:=eez;
eem:=ez;
uu=eu + ez;
ul=uu;
lu=el+ez;
ll=lu;
73
:=ArcCos[- (e/2.0)];
Return[(t1)]
];
tdown[en_,eez_]:=Module[{},
e:=en;
eu=0.1;
el=-0.1;
ez:=eez;
eem:=ez;
ul=eu - ez;
ul=uu;
ll=el-ez;
ll=lu;
:=ArcCos[- (e/2.0)];
Return[(t1)]
];
Manipulate[Plot[{Abs[tup[ee,eez]]^2 },{ee,-0.5,0.5},AxesFalse,FrameTrue,
PlotRange{-1.2,1.2},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},
BaseStyle{"Times",20},ImageSize{500,500}],{eez,0,1.0}]
Manipulate[Plot[{Abs[tdown[ee,eez]]^2 },{ee,-0.5,0.5},AxesFalse,FrameTrue,
PlotRange{-1.2,1.2},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},
BaseStyle{"Times",20},ImageSize{500,500}],{eez,0,1.0}]
(*Below I define new functions that will eventually lead to "deltapol", which
is the
differential spin polarization for the system.*)
tup1[en_,eez_]:={Abs[tup[en,eez]]^2 };
tdown1[en_,eez_]:={Abs[tdown[en,eez]]^2 };
polgen[ee_,eez_]:=Abs[(tup1[ee,eez]tdown1[ee,eez])/(tup1[ee,eez]+tdown1[ee,eez])];
polup[ee_,eez_]:=tup1[ee,eez]*polgen[ee,eez];
poldown[ee_,eez_]:=tdown1[ee,eez]*polgen[ee,eez];
deltapol[ee_,eez_]:=polup[ee,eez]-poldown[ee,eez];
Plot[polup[ee],{ee,-0.5,0.5},AxesFalse,FrameTrue,
PlotRange{-1.2,1.2},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},
BaseStyle{"Times",20},ImageSize{500,500}]
Plot[poldown[ee],{ee,-0.5,0.5},AxesFalse,FrameTrue,
PlotRange{-1.2,1.2},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},
BaseStyle{"Times",20},ImageSize{500,500}]
Plot[poldown[ee],{ee,-2.0,2.0},AxesFalse,FrameTrue,
PlotRange{-1.2,1.2},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},
BaseStyle{"Times",20},ImageSize{500,500}]
Manipulate[Plot[deltapol[ee,ezz],{ee,-2.0,2.0},AxesFalse,FrameTrue,
74
PlotRange{-1.2,1.2},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},GridLines{{},{{-1.0,Dashed},{1.0,Dashed}}},
BaseStyle{"Times",20},ImageSize{500,500}],{ezz,0,1.0}]
Needs["PlotLegends`"]
ShowLegend[ContourPlot[deltapol[ee,eez],{ee,1.2,1.2},{eez,0,0.6},PlotPoints100,Contours15,PlotRange{1.,1.},FrameTicks{Automatic,Automatic,None,None},ImageSize{500,500},FrameLabe
l{"E","EZ"},PlotLabel"Transmission",BaseStyle{"Times",20}],{ColorData["LakeC
olors"][1-#1]&,10,"
1","
1",LegendLabelNone,LegendShadowNone,LegendPosition{1.,-.7}}]
75
Appendix B:
Spin Polarizer with Two Quantum Dots
Clear["Global`*"]
v0=1.0; v=0.2;
ea=1.5;eb=-ea;e1=ea;
M:={{-v0*Exp[-I*],0,v,v},
{v,v*Exp[I*],e-ea,0},
{v,v*Exp[I*],0,e-eb},
{0,(v0*Exp[2*I*]+(e-e1)*Exp[I*]),v,v}}
MatrixForm[M]
v0=1.0
MatrixForm[M]
Mc:=Inverse[M];
const:={v0*Exp[I*],-v,-v,0};
u:=Mc.const;
t1=Part[u,2];
r1=Part[u,1];
T1[en_]:=Module[{},
e:=en;
:=ArcCos[- ((e/v0)/2.0)];
t:=t1;
Return[(t)]
];
R1[en_]:=Module[{},
e:=en;
:=ArcCos[- ((e/v0)/2.0)];
t:=r1;
Return[(t)]
];
Plot[{Abs[T1[ee]]^2 },{ee,0-2.0,2.0},AxesFalse,FrameTrue,
PlotRange{0.0,1.2},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},
BaseStyle{"Times",20}]
Plot[{Abs[R1[ee]]^2 },{ee,-2.,2.},AxesFalse,FrameTrue,
PlotRange{0.0,1.0},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},
BaseStyle{"Times",20}]
Plot[{Abs[T1[ee]]^2 +Abs[R1[ee]]^2 },{ee,-2.,2.},AxesFalse,FrameTrue,
PlotRange{0.0,1.1},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},
BaseStyle{"Times",20}]
Manipulate[Plot[{Abs[T1[ee,ee1,vv,eea]]^2 },{ee,-2.,2.},AxesFalse,FrameTrue,
PlotRange{0.0,1.2},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},GridLines{{},{{1.1,
Dashed},{1.0,Dashed},{0.5,Dashed}}},
BaseStyle{"Times",20}],{ee1,0,4},{vv,0,2},{eea,-2,2}]
76
Appendix C:
Spin Polarizer with Three Quantum Dots
Clear["Global`*"]
M:={{-v0*Exp[-I*],0,v,v,0},
{v,0,e-ea,0,v},
{v,0,0,e-eb,v},
{0,v0*Exp[2*I*],v,v,e-e1},
{0,(v0*Exp[3*I*]+(e-e2)*Exp[2*I*]),0,0,v0}}
const:={v0*Exp[I*],-v,-v,0,0};
MatrixForm[M]
MatrixForm[const]
v0=1.0;v=0.2;
e1=1.5;e2=1.5;
Mc:=Inverse[M];
u:=Mc.const;
t1=Part[u,2];
tup[en_,eez_]:=Module[{},
e:=en;
ea=eb=eez;
:=ArcCos[- ((e/v0)/2.0)];
t:=t1;
Return[(t1)]
];
tdown[en_,eez_]:=Module[{},
e:=en;
ea=eb=-eez;
:=ArcCos[- ((e/v0)/2.0)];
t:=t1;
Return[(t)]
];
tup1[en_,eez_]:={Abs[tup[en,eez]]^2 };
tdown1[en_,eez_]:={Abs[tdown[en,eez]]^2 };
polgen[ee_,eez_]:=Abs[(tup1[ee,eez]tdown1[ee,eez])/(tup1[ee,eez]+tdown1[ee,eez])];
polup[ee_,eez_]:=tup1[ee,eez]*polgen[ee,eez];
poldown[ee_,eez_]:=tdown1[ee,eez]*polgen[ee,eez];
deltapol[ee_,eez_]:=polup[ee,eez]-poldown[ee,eez];
Manipulate[Plot[deltapol[ee,ezz],{ee,-2.0,2.0},AxesFalse,FrameTrue,
PlotRange{-1.2,1.2},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},GridLines{{},{{-1.0,Dashed},{1.0,Dashed}}},
BaseStyle{"Times",20},ImageSize{500,500}],{ezz,0,1.0}]
77
Appendix D:
Program for a Single Aharonov-Bohm Ring
Clear["Global`*"]
e1=0.01;e2=-0.01;em=0.;
v0=1.0;
v1=0.1;v3=0.1;v4=0.1;v2=0.1
e1u:=e1+ez;e1d:=e1-ez;
e2u:=e2+ez;e2d:=e2-ez;
M:={{-v0*Exp[-I*],0,v1,v1,v2,v2},{v1,v3*Exp[I*],ee1u,0,0,0},{v1,v3*Exp[I*],0,e-e1d,0,0},{v2,v4*Exp[I*],0,0,ee2u,0},{v2,v4*Exp[I*],0,0,0,e-e2d},{0,-v0,v3,v3,v4,v4}}
(*v0:=1.;
e:=-2*v0*Cos[];*)
MatrixForm[M]
Mc:=Inverse[M];
const:={v0*Exp[I*],-v1,-v1,-v2,-v2,0};
u:=Mc.const;
t1=Part[u,2];
r1=Part[u,1];
T1[en_]:=Module[{},
e:=en;
ez:=0.08;
:=ArcCos[- ((e/v0)/2.0)];
t:=t1;
Return[(t)]
];
R1[en_]:=Module[{},
e:=en;
:=ArcCos[- ((e/v0)/2.0)];
t:=r1;
Return[(t)]
];
Plot[{Abs[T1[ee]]^2 },{ee,0-0.5,0.5},AxesFalse,FrameTrue,
PlotRange{0.0,1.0},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},
BaseStyle{"Times",20}]
Plot[{Abs[R1[ee]]^2 },{ee,-2.,2.},AxesFalse,FrameTrue,
PlotRange{0.0,1.0},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},
BaseStyle{"Times",20}]
Plot[{Abs[T1[ee]]^2 +Abs[R1[ee]]^2 },{ee,-2.,2.},AxesFalse,FrameTrue,
PlotRange{0.0,1.1},PlotStyle{AbsoluteThickness[2.8]},
FrameLabel {"E","T"},
BaseStyle{"Times",20}]
78
Appendix E:
Routine for Calculating IV Curves of Nanodevices using Simpson’s Rule
(*This routine calculates the theoretical current for the devices. It uses
simpson’s rule to integrate the transmission of the system. It relates the
Fermi energy of the system and it’s electron transmission to applied voltages,
and finds the current from this integration [28]. Credit goes to Sadeq
Malakooti for building this code.*)
(*The variables are defined as follows:
-ef is the Fermi energy of the leads. These need to be set to zero, so that the
dispersion relation holds for the transmission/reflection calculations.
-k is Boltzman’s constant in ev/T.
-q is the ambient temperature around the device in Kelvin.
- is just the inverse of boltzman’s constant times the temperature.
-is a parameter which controls how the voltage is applied to the system. If
the voltage is applied evenly to the system (half at one end, the rest at the
other end), then it is equal to 0.5.
-v is the voltage being applied across the system*)
ef:=0.0;
k:=8.6174*10^(-5);
q:=300;
:=1/(k*q);
:=1/2;
IV=Function[e,(Abs[T[e]]^2)*(1/(Exp[*{e-ef-(1-)*v}]+1)-1/(Exp[*{eef+(*v)}]+1))];
Simpson1[a0_,b0_,m0_]:=
Module[{a=N[a0],b=N[b0],m=m0,k},
h=(b-a)/(2m);
SumEven=0;
For[k=1,km-1,k++,
SumEven=SumEven+IV[a+ h 2 k];];
SumOdd=0;
For[k=1,km,k++,
SumOdd=SumOdd+IV[a+h (2 k-1)];];
Return[h/3 (IV[a]+IV[b]+2 SumEven+4 SumOdd)];];
current=Simpson1[-2.0, 2.0,100];
Plot[current,{v,-5,5},AxesFalse,FrameTrue,FrameStyleThick,FrameLabel{"Vsd
(Volts)","I(A)"},BaseStyle{"Helvetica",22,Bold},PlotStyle{AbsoluteThickness[
3]}, PlotRange{{-1,1}},ImageSize{500,500}]